Global wellposedness for the dissipative system modeling electrohydrodynamics with large vertical velocity component in critical Besov space
Jihong Zhao  College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China (email) Abstract: In this paper, we are concerned with a model arising from electrohydrodynamics, which is a coupled system of the NavierStokes equations and the PoissonNernstPlanck equations through charge transport and external forcing terms. The local wellposedness and global wellposedness with small initial data to the 3D Cauchy problem of this system are established in the critical Besov space $\dot{B}^{1+\frac{3}{p}}_{p,1}(\mathbb{R}^{3})\times(\dot{B}^{2+\frac{3}{q}}_{q,1}(\mathbb{R}^{3}))^{2}$ with suitable choices of $p, q$. Especially, we prove that there exist two positive constants $c_{0}, C_{0}$ depending on the coefficients of system except $\mu$ such that if \begin{equation*} \big(\u_{0}^{h}\_{\dot{B}^{1+\frac{3}{p}}_{p,1}}+(\mu+1)\(v_{0},w_{0})\_{\dot{B}^{2+\frac{3}{q}}_{q,1}} \big) \exp\Big\{\frac{C_{0}}{\mu^{2}}(\u_{0}^{3}\_{\dot{B}^{1+\frac{3}{p}}_{p,1}}^{2}+1)\Big\}\leq c_{0}\mu, \end{equation*} then the above local solution can be extended to the global one. This result implies the global wellposedness of this system with large initial vertical velocity component.
Keywords: NavierStokes equations, PoissonNernstPlanck equations, electrohydrodynamics, wellposedness, Besov space.
Received: November 2013; Revised: June 2014; Available Online: August 2014. 
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